The symmetry of rotating fluid bodies

Consider an incompressible fluid body (in outer space) rotating about an axis with a given angular velocity , and which is in equilibrium relative to the potential energy of its own gravitational field and the surface energy due to surface tension. We show that such a body possesses a plane of symmetry perpendicular to the axis of rotation such that any line parallel to the axis and meeting the body cuts it in a line segment whose center lies on the plane of symmetry. This extends an earlier result of L. Lichtenstein [4].This work was done with the support of S.F.B. 72 while the author was a visitor at the University of Bonn